src/HOL/Bali/AxExample.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 62390 842917225d56 child 67399 eab6ce8368fa permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Bali/AxExample.thy
```
```     2     Author:     David von Oheimb
```
```     3 *)
```
```     4
```
```     5 subsection \<open>Example of a proof based on the Bali axiomatic semantics\<close>
```
```     6
```
```     7 theory AxExample
```
```     8 imports AxSem Example
```
```     9 begin
```
```    10
```
```    11 definition
```
```    12   arr_inv :: "st \<Rightarrow> bool" where
```
```    13  "arr_inv = (\<lambda>s. \<exists>obj a T el. globs s (Stat Base) = Some obj \<and>
```
```    14                               values obj (Inl (arr, Base)) = Some (Addr a) \<and>
```
```    15                               heap s a = Some \<lparr>tag=Arr T 2,values=el\<rparr>)"
```
```    16
```
```    17 lemma arr_inv_new_obj:
```
```    18 "\<And>a. \<lbrakk>arr_inv s; new_Addr (heap s)=Some a\<rbrakk> \<Longrightarrow> arr_inv (gupd(Inl a\<mapsto>x) s)"
```
```    19 apply (unfold arr_inv_def)
```
```    20 apply (force dest!: new_AddrD2)
```
```    21 done
```
```    22
```
```    23 lemma arr_inv_set_locals [simp]: "arr_inv (set_locals l s) = arr_inv s"
```
```    24 apply (unfold arr_inv_def)
```
```    25 apply (simp (no_asm))
```
```    26 done
```
```    27
```
```    28 lemma arr_inv_gupd_Stat [simp]:
```
```    29   "Base \<noteq> C \<Longrightarrow> arr_inv (gupd(Stat C\<mapsto>obj) s) = arr_inv s"
```
```    30 apply (unfold arr_inv_def)
```
```    31 apply (simp (no_asm_simp))
```
```    32 done
```
```    33
```
```    34 lemma ax_inv_lupd [simp]: "arr_inv (lupd(x\<mapsto>y) s) = arr_inv s"
```
```    35 apply (unfold arr_inv_def)
```
```    36 apply (simp (no_asm))
```
```    37 done
```
```    38
```
```    39
```
```    40 declare if_split_asm [split del]
```
```    41 declare lvar_def [simp]
```
```    42
```
```    43 ML \<open>
```
```    44 fun inst1_tac ctxt s t xs st =
```
```    45   (case AList.lookup (op =) (rev (Term.add_var_names (Thm.prop_of st) [])) s of
```
```    46     SOME i => PRIMITIVE (Rule_Insts.read_instantiate ctxt [(((s, i), Position.none), t)] xs) st
```
```    47   | NONE => Seq.empty);
```
```    48
```
```    49 fun ax_tac ctxt =
```
```    50   REPEAT o resolve_tac ctxt [allI] THEN'
```
```    51   resolve_tac ctxt
```
```    52     @{thms ax_Skip ax_StatRef ax_MethdN ax_Alloc ax_Alloc_Arr ax_SXAlloc_Normal ax_derivs.intros(8-)};
```
```    53 \<close>
```
```    54
```
```    55
```
```    56 theorem ax_test: "tprg,({}::'a triple set)\<turnstile>
```
```    57   {Normal (\<lambda>Y s Z::'a. heap_free four s \<and> \<not>initd Base s \<and> \<not> initd Ext s)}
```
```    58   .test [Class Base].
```
```    59   {\<lambda>Y s Z. abrupt s = Some (Xcpt (Std IndOutBound))}"
```
```    60 apply (unfold test_def arr_viewed_from_def)
```
```    61 apply (tactic "ax_tac @{context} 1" (*;;*))
```
```    62 defer (* We begin with the last assertion, to synthesise the intermediate
```
```    63          assertions, like in the fashion of the weakest
```
```    64          precondition. *)
```
```    65 apply  (tactic "ax_tac @{context} 1" (* Try *))
```
```    66 defer
```
```    67 apply    (tactic \<open>inst1_tac @{context} "Q"
```
```    68                  "\<lambda>Y s Z. arr_inv (snd s) \<and> tprg,s\<turnstile>catch SXcpt NullPointer" []\<close>)
```
```    69 prefer 2
```
```    70 apply    simp
```
```    71 apply   (rule_tac P' = "Normal (\<lambda>Y s Z. arr_inv (snd s))" in conseq1)
```
```    72 prefer 2
```
```    73 apply    clarsimp
```
```    74 apply   (rule_tac Q' = "(\<lambda>Y s Z. Q Y s Z)\<leftarrow>=False\<down>=\<diamondsuit>" and Q = Q for Q in conseq2)
```
```    75 prefer 2
```
```    76 apply    simp
```
```    77 apply   (tactic "ax_tac @{context} 1" (* While *))
```
```    78 prefer 2
```
```    79 apply    (rule ax_impossible [THEN conseq1], clarsimp)
```
```    80 apply   (rule_tac P' = "Normal P" and P = P for P in conseq1)
```
```    81 prefer 2
```
```    82 apply    clarsimp
```
```    83 apply   (tactic "ax_tac @{context} 1")
```
```    84 apply   (tactic "ax_tac @{context} 1" (* AVar *))
```
```    85 prefer 2
```
```    86 apply    (rule ax_subst_Val_allI)
```
```    87 apply    (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a. Normal (PP a\<leftarrow>x)" ["PP", "x"]\<close>)
```
```    88 apply    (simp del: avar_def2 peek_and_def2)
```
```    89 apply    (tactic "ax_tac @{context} 1")
```
```    90 apply   (tactic "ax_tac @{context} 1")
```
```    91       (* just for clarification: *)
```
```    92 apply   (rule_tac Q' = "Normal (\<lambda>Var:(v, f) u ua. fst (snd (avar tprg (Intg 2) v u)) = Some (Xcpt (Std IndOutBound)))" in conseq2)
```
```    93 prefer 2
```
```    94 apply    (clarsimp simp add: split_beta)
```
```    95 apply   (tactic "ax_tac @{context} 1" (* FVar *))
```
```    96 apply    (tactic "ax_tac @{context} 2" (* StatRef *))
```
```    97 apply   (rule ax_derivs.Done [THEN conseq1])
```
```    98 apply   (clarsimp simp add: arr_inv_def inited_def in_bounds_def)
```
```    99 defer
```
```   100 apply  (rule ax_SXAlloc_catch_SXcpt)
```
```   101 apply  (rule_tac Q' = "(\<lambda>Y (x, s) Z. x = Some (Xcpt (Std NullPointer)) \<and> arr_inv s) \<and>. heap_free two" in conseq2)
```
```   102 prefer 2
```
```   103 apply   (simp add: arr_inv_new_obj)
```
```   104 apply  (tactic "ax_tac @{context} 1")
```
```   105 apply  (rule_tac C = "Ext" in ax_Call_known_DynT)
```
```   106 apply     (unfold DynT_prop_def)
```
```   107 apply     (simp (no_asm))
```
```   108 apply    (intro strip)
```
```   109 apply    (rule_tac P' = "Normal P" and P = P for P in conseq1)
```
```   110 apply     (tactic "ax_tac @{context} 1" (* Methd *))
```
```   111 apply     (rule ax_thin [OF _ empty_subsetI])
```
```   112 apply     (simp (no_asm) add: body_def2)
```
```   113 apply     (tactic "ax_tac @{context} 1" (* Body *))
```
```   114 (* apply       (rule_tac [2] ax_derivs.Abrupt) *)
```
```   115 defer
```
```   116 apply      (simp (no_asm))
```
```   117 apply      (tactic "ax_tac @{context} 1") (* Comp *)
```
```   118             (* The first statement in the  composition
```
```   119                  ((Ext)z).vee = 1; Return Null
```
```   120                 will throw an exception (since z is null). So we can handle
```
```   121                 Return Null with the Abrupt rule *)
```
```   122 apply       (rule_tac [2] ax_derivs.Abrupt)
```
```   123
```
```   124 apply      (rule ax_derivs.Expr) (* Expr *)
```
```   125 apply      (tactic "ax_tac @{context} 1") (* Ass *)
```
```   126 prefer 2
```
```   127 apply       (rule ax_subst_Var_allI)
```
```   128 apply       (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a vs l vf. PP a vs l vf\<leftarrow>x \<and>. p" ["PP", "x", "p"]\<close>)
```
```   129 apply       (rule allI)
```
```   130 apply       (tactic \<open>simp_tac (@{context} delloop "split_all_tac" delsimps [@{thm peek_and_def2}, @{thm heap_def2}, @{thm subst_res_def2}, @{thm normal_def2}]) 1\<close>)
```
```   131 apply       (rule ax_derivs.Abrupt)
```
```   132 apply      (simp (no_asm))
```
```   133 apply      (tactic "ax_tac @{context} 1" (* FVar *))
```
```   134 apply       (tactic "ax_tac @{context} 2", tactic "ax_tac @{context} 2", tactic "ax_tac @{context} 2")
```
```   135 apply      (tactic "ax_tac @{context} 1")
```
```   136 apply     (tactic \<open>inst1_tac @{context} "R" "\<lambda>a'. Normal ((\<lambda>Vals:vs (x, s) Z. arr_inv s \<and> inited Ext (globs s) \<and> a' \<noteq> Null \<and> vs = [Null]) \<and>. heap_free two)" []\<close>)
```
```   137 apply     fastforce
```
```   138 prefer 4
```
```   139 apply    (rule ax_derivs.Done [THEN conseq1],force)
```
```   140 apply   (rule ax_subst_Val_allI)
```
```   141 apply   (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a. Normal (PP a\<leftarrow>x)" ["PP", "x"]\<close>)
```
```   142 apply   (simp (no_asm) del: peek_and_def2 heap_free_def2 normal_def2 o_apply)
```
```   143 apply   (tactic "ax_tac @{context} 1")
```
```   144 prefer 2
```
```   145 apply   (rule ax_subst_Val_allI)
```
```   146 apply    (tactic \<open>inst1_tac @{context} "P'" "\<lambda>aa v. Normal (QQ aa v\<leftarrow>y)" ["QQ", "y"]\<close>)
```
```   147 apply    (simp del: peek_and_def2 heap_free_def2 normal_def2)
```
```   148 apply    (tactic "ax_tac @{context} 1")
```
```   149 apply   (tactic "ax_tac @{context} 1")
```
```   150 apply  (tactic "ax_tac @{context} 1")
```
```   151 apply  (tactic "ax_tac @{context} 1")
```
```   152 (* end method call *)
```
```   153 apply (simp (no_asm))
```
```   154     (* just for clarification: *)
```
```   155 apply (rule_tac Q' = "Normal ((\<lambda>Y (x, s) Z. arr_inv s \<and> (\<exists>a. the (locals s (VName e)) = Addr a \<and> obj_class (the (globs s (Inl a))) = Ext \<and>
```
```   156  invocation_declclass tprg IntVir s (the (locals s (VName e))) (ClassT Base)
```
```   157      \<lparr>name = foo, parTs = [Class Base]\<rparr> = Ext)) \<and>. initd Ext \<and>. heap_free two)"
```
```   158   in conseq2)
```
```   159 prefer 2
```
```   160 apply  clarsimp
```
```   161 apply (tactic "ax_tac @{context} 1")
```
```   162 apply (tactic "ax_tac @{context} 1")
```
```   163 defer
```
```   164 apply  (rule ax_subst_Var_allI)
```
```   165 apply  (tactic \<open>inst1_tac @{context} "P'" "\<lambda>vf. Normal (PP vf \<and>. p)" ["PP", "p"]\<close>)
```
```   166 apply  (simp (no_asm) del: split_paired_All peek_and_def2 initd_def2 heap_free_def2 normal_def2)
```
```   167 apply  (tactic "ax_tac @{context} 1" (* NewC *))
```
```   168 apply  (tactic "ax_tac @{context} 1" (* ax_Alloc *))
```
```   169      (* just for clarification: *)
```
```   170 apply  (rule_tac Q' = "Normal ((\<lambda>Y s Z. arr_inv (store s) \<and> vf=lvar (VName e) (store s)) \<and>. heap_free three \<and>. initd Ext)" in conseq2)
```
```   171 prefer 2
```
```   172 apply   (simp add: invocation_declclass_def dynmethd_def)
```
```   173 apply   (unfold dynlookup_def)
```
```   174 apply   (simp add: dynmethd_Ext_foo)
```
```   175 apply   (force elim!: arr_inv_new_obj atleast_free_SucD atleast_free_weaken)
```
```   176      (* begin init *)
```
```   177 apply  (rule ax_InitS)
```
```   178 apply     force
```
```   179 apply    (simp (no_asm))
```
```   180 apply   (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
```
```   181 apply   (rule ax_Init_Skip_lemma)
```
```   182 apply  (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
```
```   183 apply  (rule ax_InitS [THEN conseq1] (* init Base *))
```
```   184 apply      force
```
```   185 apply     (simp (no_asm))
```
```   186 apply    (unfold arr_viewed_from_def)
```
```   187 apply    (rule allI)
```
```   188 apply    (rule_tac P' = "Normal P" and P = P for P in conseq1)
```
```   189 apply     (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
```
```   190 apply     (tactic "ax_tac @{context} 1")
```
```   191 apply     (tactic "ax_tac @{context} 1")
```
```   192 apply     (rule_tac [2] ax_subst_Var_allI)
```
```   193 apply      (tactic \<open>inst1_tac @{context} "P'" "\<lambda>vf l vfa. Normal (P vf l vfa)" ["P"]\<close>)
```
```   194 apply     (tactic \<open>simp_tac (@{context} delloop "split_all_tac" delsimps [@{thm split_paired_All}, @{thm peek_and_def2}, @{thm heap_free_def2}, @{thm initd_def2}, @{thm normal_def2}, @{thm supd_lupd}]) 2\<close>)
```
```   195 apply      (tactic "ax_tac @{context} 2" (* NewA *))
```
```   196 apply       (tactic "ax_tac @{context} 3" (* ax_Alloc_Arr *))
```
```   197 apply       (tactic "ax_tac @{context} 3")
```
```   198 apply      (tactic \<open>inst1_tac @{context} "P" "\<lambda>vf l vfa. Normal (P vf l vfa\<leftarrow>\<diamondsuit>)" ["P"]\<close>)
```
```   199 apply      (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 2\<close>)
```
```   200 apply      (tactic "ax_tac @{context} 2")
```
```   201 apply     (tactic "ax_tac @{context} 1" (* FVar *))
```
```   202 apply      (tactic "ax_tac @{context} 2" (* StatRef *))
```
```   203 apply     (rule ax_derivs.Done [THEN conseq1])
```
```   204 apply     (tactic \<open>inst1_tac @{context} "Q" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf=lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Base \<and>. initd Ext)" []\<close>)
```
```   205 apply     (clarsimp split del: if_split)
```
```   206 apply     (frule atleast_free_weaken [THEN atleast_free_weaken])
```
```   207 apply     (drule initedD)
```
```   208 apply     (clarsimp elim!: atleast_free_SucD simp add: arr_inv_def)
```
```   209 apply    force
```
```   210 apply   (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
```
```   211 apply   (rule ax_triv_Init_Object [THEN peek_and_forget2, THEN conseq1])
```
```   212 apply     (rule wf_tprg)
```
```   213 apply    clarsimp
```
```   214 apply   (tactic \<open>inst1_tac @{context} "P" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Ext)" []\<close>)
```
```   215 apply   clarsimp
```
```   216 apply  (tactic \<open>inst1_tac @{context} "PP" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. Not \<circ> initd Base)" []\<close>)
```
```   217 apply  clarsimp
```
```   218      (* end init *)
```
```   219 apply (rule conseq1)
```
```   220 apply (tactic "ax_tac @{context} 1")
```
```   221 apply clarsimp
```
```   222 done
```
```   223
```
```   224 (*
```
```   225 while (true) {
```
```   226   if (i) {throw xcpt;}
```
```   227   else i=j
```
```   228 }
```
```   229 *)
```
```   230 lemma Loop_Xcpt_benchmark:
```
```   231  "Q = (\<lambda>Y (x,s) Z. x \<noteq> None \<longrightarrow> the_Bool (the (locals s i))) \<Longrightarrow>
```
```   232   G,({}::'a triple set)\<turnstile>{Normal (\<lambda>Y s Z::'a. True)}
```
```   233   .lab1\<bullet> While(Lit (Bool True)) (If(Acc (LVar i)) (Throw (Acc (LVar xcpt))) Else
```
```   234         (Expr (Ass (LVar i) (Acc (LVar j))))). {Q}"
```
```   235 apply (rule_tac P' = "Q" and Q' = "Q\<leftarrow>=False\<down>=\<diamondsuit>" in conseq12)
```
```   236 apply  safe
```
```   237 apply  (tactic "ax_tac @{context} 1" (* Loop *))
```
```   238 apply   (rule ax_Normal_cases)
```
```   239 prefer 2
```
```   240 apply    (rule ax_derivs.Abrupt [THEN conseq1], clarsimp simp add: Let_def)
```
```   241 apply   (rule conseq1)
```
```   242 apply    (tactic "ax_tac @{context} 1")
```
```   243 apply   clarsimp
```
```   244 prefer 2
```
```   245 apply  clarsimp
```
```   246 apply (tactic "ax_tac @{context} 1" (* If *))
```
```   247 apply  (tactic
```
```   248   \<open>inst1_tac @{context} "P'" "Normal (\<lambda>s.. (\<lambda>Y s Z. True)\<down>=Val (the (locals s i)))" []\<close>)
```
```   249 apply  (tactic "ax_tac @{context} 1")
```
```   250 apply  (rule conseq1)
```
```   251 apply   (tactic "ax_tac @{context} 1")
```
```   252 apply  clarsimp
```
```   253 apply (rule allI)
```
```   254 apply (rule ax_escape)
```
```   255 apply auto
```
```   256 apply  (rule conseq1)
```
```   257 apply   (tactic "ax_tac @{context} 1" (* Throw *))
```
```   258 apply   (tactic "ax_tac @{context} 1")
```
```   259 apply   (tactic "ax_tac @{context} 1")
```
```   260 apply  clarsimp
```
```   261 apply (rule_tac Q' = "Normal (\<lambda>Y s Z. True)" in conseq2)
```
```   262 prefer 2
```
```   263 apply  clarsimp
```
```   264 apply (rule conseq1)
```
```   265 apply  (tactic "ax_tac @{context} 1")
```
```   266 apply  (tactic "ax_tac @{context} 1")
```
```   267 prefer 2
```
```   268 apply   (rule ax_subst_Var_allI)
```
```   269 apply   (tactic \<open>inst1_tac @{context} "P'" "\<lambda>b Y ba Z vf. \<lambda>Y (x,s) Z. x=None \<and> snd vf = snd (lvar i s)" []\<close>)
```
```   270 apply   (rule allI)
```
```   271 apply   (rule_tac P' = "Normal P" and P = P for P in conseq1)
```
```   272 prefer 2
```
```   273 apply    clarsimp
```
```   274 apply   (tactic "ax_tac @{context} 1")
```
```   275 apply   (rule conseq1)
```
```   276 apply    (tactic "ax_tac @{context} 1")
```
```   277 apply   clarsimp
```
```   278 apply  (tactic "ax_tac @{context} 1")
```
```   279 apply clarsimp
```
```   280 done
```
```   281
```
```   282 end
```
```   283
```